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It is well known that if $f:\mathbb R\to \mathbb R$ is real analytic then we have that either $f\equiv 0$ or $f$ has finitely many zeros on the interval $[a,b]$.

It is also known that you can have a nonnegative function $f:\mathbb R\to \mathbb R$ that is smooth with infinitely many zeros on $[a,b]$. A bump function, for example.

My question is:

Given a nonnegative function $f\in C^1(\mathbb R,\mathbb R)$, not identically $0$, is there an additional constraint other than analytic that tells you that $f$ has finitely many zeros on the interval $[a,b]$?

It is well known that if $f:\mathbb R\to \mathbb R$ is real analytic then we have that either $f\equiv 0$ or $f$ has finitely many zeros on the interval $[a,b]$.

It is also known that you can have a nonnegative function $f:\mathbb R\to \mathbb R$ that is smooth with infinitely many zeros on $[a,b]$. A bump function, for example.

My question is:

Given a nonnegative function $f\in C^1([a,b],\mathbb R)$, not identically $0$, is there an additional constraint other than analytic that tells you that $f$ has finitely many zeros?

It is well known that if $f:\mathbb R\to \mathbb R$ is real analytic then we have that either $f\equiv 0$ or $f$ has finitely many zeros on the interval $[a,b]$.

It is also known that you can have a nonnegative function $f:\mathbb R\to \mathbb R$ that is smooth with infinitely many zeros on $[a,b]$. A bump function, for example.

My question is:

Given a nonnegative function $f\in C^1([a,b],\mathbb R)$, not identically $0$, is there an additional constraint other than analytic that tells you that $f$ has finitely many zeros?

It is well known that if $f:\mathbb R\to \mathbb R$ is real analytic then we have that either $f\equiv 0$ or $f$ has finitely many zeros on the interval $[a,b]$.

It is also known that you can have a nonnegative function $f:\mathbb R\to \mathbb R$ that is smooth with infinitely many zeros on $[a,b]$. A bump function, for example.

My question is:

Given a nonnegative function $f\in C^1(\mathbb R,\mathbb R)$, not identically $0$, is there an additional constraint other than analytic that tells you that $f$ has finitely many zeros on the interval $[a,b]$?

It is well known that if $f:[a,b]\to \mathbb R$ is real analytic then we have that either $f\equiv 0$ or $f$ has finitely many zeros.

It is also known that you can have a nonnegative function $f:[a,b]\to \mathbb R$ that is smooth with infinitely many zeros. A bump function, for example.

My question is:

Given a nonnegative function $f\in C^1([a,b],\mathbb R)$, not identically $0$, is there an additional constraint other than analytic that tells you that $f$ has finitely many zeros?

It is well known that if $f:\mathbb R\to \mathbb R$ is real analytic then we have that either $f\equiv 0$ or $f$ has finitely many zeros on the interval $[a,b]$.

It is also known that you can have a nonnegative function $f:\mathbb R\to \mathbb R$ that is smooth with infinitely many zeros on $[a,b]$. A bump function, for example.

My question is:

Given a nonnegative function $f\in C^1([a,b],\mathbb R)$, not identically $0$, is there an additional constraint other than analytic that tells you that $f$ has finitely many zeros?